From 7dfcc480ba1e19bd3232349fc733caef94034292 Mon Sep 17 00:00:00 2001 From: stainer_t Date: Mon, 8 Sep 2025 13:48:49 +0200 Subject: Initial commit from Polytechnique Montreal --- doc/IGE344/SectDOPTIMIZE.tex | 276 +++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 276 insertions(+) create mode 100644 doc/IGE344/SectDOPTIMIZE.tex (limited to 'doc/IGE344/SectDOPTIMIZE.tex') diff --git a/doc/IGE344/SectDOPTIMIZE.tex b/doc/IGE344/SectDOPTIMIZE.tex new file mode 100644 index 0000000..f07ff88 --- /dev/null +++ b/doc/IGE344/SectDOPTIMIZE.tex @@ -0,0 +1,276 @@ +\subsection{Contents of a \dir{optimize} data structure} + +The \dir{optimize} specification is used to store the optimization variables and functions values and definitions, limits and +options. + +\vskip 0.08cm + +In any case, the signature variable for this data structure must be $\mathsf{SIGNA}$=\verb*|L_OPTIMIZE |. The dimensioning +parameters for this data structure, which are stored in the state vector $\mathcal{S}^{o}_{i}$, represents: + +\begin{itemize} +\item The number of decision variables $N_{var} = \mathcal{S}^{o}_{1}$. +\item The number of constraints $N_{cst} = \mathcal{S}^{o}_{2}$. +\item The type of optimization $\mathcal{S}^{o}_{3}$, where + +\begin{displaymath} +\mathcal{S}^{o}_{3} = \left\{ +\begin{array}{rl} + 1 & \textrm{minimization} \\ + -1 & \textrm{maximization} \\ +\end{array} \right. +\end{displaymath} + +\item The result of a test for external convergence of the optimization problem $\mathcal{S}^{o}_{4}$, where + +\begin{displaymath} +\mathcal{S}^{o}_{4} = \left\{ +\begin{array}{rl} + 0 & \textrm{not converged} \\ + 1 & \textrm{converged} \\ +\end{array} \right. +\end{displaymath} + +\item The index of external iteration (${S}^{o}_{5}$). If module {\tt PLQ:} is used, this is the iteration index relative to +the solution of a new linear optimization problem with a quadratic constraint. If module {\tt LNSR:} is used, this is the number +of times the line search algorithm is called. + +\item The type of reduction for the radius if the quadratic constraint ($\mathcal{S}^{o}_{6}$) used in module {\tt PLQ:}, where + +\begin{displaymath} +\mathcal{S}^{o}_{6} = \left\{ +\begin{array}{rl} + 1 & \textrm{half} \\ + 2 & \textrm{parabolic} \\ +\end{array} \right. +\end{displaymath} + +\item The type of gradient search $\mathcal{S}^{o}_{7}$, where + +\begin{displaymath} +\mathcal{S}^{o}_{7} = \left\{ +\begin{array}{rl} + 0 & \textrm{steepest descent} \\ + 1 & \textrm{conjugate gradient} \\ + 2 & \textrm{Broyden-Fletcher-Goldfarb-Shanno (BFGS)} \\ + 3 & \textrm{memory limited Broyden-Fletcher-Goldfarb-Shanno (LBFGS)} \\ + 4 & \textrm{Newton method for unconstrained optimization} \\ +\end{array} \right. +\end{displaymath} + +\item The type of optimization method $\mathcal{S}^{o}_{8}$, where + +\begin{displaymath} +\mathcal{S}^{o}_{8} = \left\{ +\begin{array}{rl} + 0 & \textrm{not set} \\ + 1 & \textrm{OPTEX method with module {\tt PLQ:}} \\ + 2 & \textrm{line search algorithm with module {\tt LNSR:}} \\ + 3 & \textrm{fixed point SPH algorithm with module {\tt FPSPH:}} \\ + 4 & \textrm{Newtonian SPH algorithm with module {\tt FPSPH:}} \\ +\end{array} \right. +\end{displaymath} + +\item The resolution's method for the linear problem with quadratic constraint used in module {\tt PLQ:} ($\mathcal{S}^{o}_{9})$, where + +\begin{displaymath} +\mathcal{S}^{o}_{9} = \left\{ +\begin{array}{rl} + 1 & \textrm{SIMPLEX/LEMKE} \\ + 2 & \textrm{LEMKE/LEMKE} \\ + 3 & \textrm{MAP} \\ + 4 & \textrm{Augmented Lagragian} \\ + 5 & \textrm{Penalty Method} \\ +\end{array} \right. +\end{displaymath} + +\item The index of line search iteration in module {\tt LNSR:} (${S}^{o}_{10}$). The maximum number of line search iterations is fixed to 50 in module {\tt LNSR:}. + +\item The result of a test for the convergence of the line search iterations $\mathcal{S}^{o}_{11}$ in module {\tt LNSR:}, where + +\begin{displaymath} +\mathcal{S}^{o}_{11} = \left\{ +\begin{array}{rl} + 0 & \textrm{not converged} \\ + 1 & \textrm{converged} \\ + 2 & \textrm{maximum line search iteration reached} \\ +\end{array} \right. +\end{displaymath} + +\item The maximum number of external iterations in module {\tt LNSR:} (${S}^{o}_{12}$). + +\item The external iteration restart cycle in module {\tt LNSR:} (${S}^{o}_{13}$). + +\item A flag for unsuccessful resolution in module {\tt PLQ:} ${S}^{o}_{14}$, where + +\begin{displaymath} +\mathcal{S}^{o}_{14} = \left\{ +\begin{array}{rl} + 0 & \textrm{successful at last iteration} \\ + \ge 1 & \textrm{number of iteration with unsuccessful resolution.} \\ +\end{array} \right. +\end{displaymath} + +\end{itemize} + +\begin{DescriptionEnregistrement}{Main records and sub-directories in \dir{optimize}}{7.0cm} + +\CharEnr +{SIGNATURE\blank{3}}{$*12$} +{Signature of the data structure ($\mathsf{SIGNA}$)} + +\IntEnr +{STATE-VECTOR}{$40$} +{Vector describing the various parameters associated with data structure $\mathcal{S}^{o}_{i}$.} + +\OptIntEnr +{DEL-STATE\blank{3}}{$40$}{*} +{Vector describing the various parameters associated with data structure $\mathcal{S}^{g}_{i}$. This array is + available if the {\sc optimize} object has been created using module {\tt DLEAK:} or {\tt DSPH:}.} + +\DbleEnr +{VAR-VALUE\blank{3}}{$N_{var}$}{} +{The values of the decision variables} + +\DbleEnr +{VAR-VAL-MAX\blank{1}}{$N_{var}$}{} +{The maximum values of the decision variables can be.} + +\DbleEnr +{VAR-VAL-MIN\blank{1}}{$N_{var}$}{} +{The minimum values of the decision variables can be.} + +\DbleEnr +{VAR-WEIGHT\blank{2}}{$N_{var}$}{} +{The weight of the decision variables $w_{i}$ in the quadratic constraint.} + +\DbleEnr +{CST-OBJ\blank{5}}{$N_{cst}$}{} +{The limit value of the contraints. The units depends with the type of the constraint type.} + +\IntEnr +{CST-TYPE\blank{4}}{$N_{cst}$} +{The type of the contraints: =-1 for $\geq$; =0 for $=$; =1 for $\leq$.} + +\DbleEnr +{CST-WEIGHT\blank{2}}{$N_{cst}$}{} +{The weight of the constraint $\eta_{j}$ and $\gamma_{j}$ for the duals and meta-heuristic methods.} + +\DbleEnr +{FOBJ-CST-VAL}{$N_{cst}+1$}{} +{The actual values of the objective function (first value) and the contraints (the following values). The +number of the constraints are assigned in the order they have been defined.} + +\DbleEnr +{OPT-PARAM-R\blank{1}}{$40$}{} +{The different limits and values for the iterative calculations of the optimization problem.} + +\DbleEnr +{GRADIENT\blank{4}}{$N_{var}, N_{cst}+1$}{} +{The gradients of the objective function and the constraints. The gradients of the objective for all the +decision variables are in first position, then follow the gradients of the constraints.} + +\OptDbleEnr +{GRADIENT-DIR}{$N_{var}, N_{cst}+1$}{*}{} +{The direct component (without the flux effect) for the gradients of the objective function and the constraints.} + +\OptDbleEnr +{DIRECTION\blank{3}}{$N_{var}$}{$\mathcal{S}^{o}_{8}=2$}{} +{Direction of the line search.} + +\OptDbleEnr +{HESSIAN\blank{5}}{$N_{var}, N_{var}$}{$\mathcal{S}^{o}_{7}=2$}{} +{The hessian matrix containing second derivatives of the objective function with respect to the decision variables.} + +\OptDbleEnr +{LNSR-INFO\blank{3}}{$9$}{$\mathcal{S}^{o}_{8}=2$}{} +{Double precision values used by module {\tt LNSR:}.} + +\DirEnr +{OLD-VALUE\blank{3}} +{Directory containing differents informations of the previous iterations. the values of the decision +variables, the objective function, the constraints and the gradients of these functions for the previous +external iteration. This repertory will be created by the module \moc{PLQ:} (unless it is specified to not do) or by module \moc{LNSR:}.} + +\end{DescriptionEnregistrement} + +The array {\tt OPT-PARAM-R} contains double precision values related with the different limits and values for the iterative calculations of the optimization problem: \\ +\begin{tabular}{p{0.06\textwidth}p{0.1\textwidth}p{0.75\textwidth}} +1st & $S$ & initial radius of the quadratic constraint in module {\tt PLQ:} or maximum stepsize for the line +search in module {\moc LNSR:} (default: 1.0).\\ +2nd & $\delta$ & initial size of the hypercube for MAP method. (default: 0.1).\\ +3rd & $\varepsilon_{\rm ext}$ & limit for external convergence (default: $10^{-4}$).\\ +4th & $\varepsilon_{\rm int}$ & limit for internal convergence (default: $10^{-4}$).\\ +5th & $\varepsilon_{\rm quad}$ & limit for convergence of the quadratic constraint in module {\moc PLQ:} (default: $10^{-4}$). \\ +\end{tabular} \\ +The other value of the record are not used and set to 0.0. + +\vskip 0.2cm + +The optional array {\tt DEL-STATE} contains integer values related to the definition of mixture and group indices in module {\tt DLEAK:}. +\begin{itemize} +\item The number of energy groups in macrolib $\mathcal{S}^{g}_{1}$. +\item The number of mixtures in macrolib $\mathcal{S}^{g}_{2}$. +\item The type of leakage parameters $\mathcal{S}^{g}_{3}$, +where +\begin{displaymath} +\mathcal{S}^{g}_{3} = \left\{ +\begin{array}{rl} +1 & \textrm{use diffusion coefficients} \\ +2 & \textrm{use $P_1$-weighted macroscopic total cross sections.} \\ +\end{array} \right. +\end{displaymath} + +\item The type of control variables $\mathcal{S}^{g}_{4}$, +where +\begin{displaymath} +\mathcal{S}^{g}_{4} = \left\{ +\begin{array}{rl} +1 & \textrm{use leakage parameters} \\ +2 & \textrm{use correction factors on leakage parameters} \\ +3 & \textrm{use SPH factors compatible with diffusion theory, $P_n$ and $SP_n$ equations} \\ +4 & \textrm{use SPH factors compatible with other types of transport-theory macro-} \\ +& \textrm{calculations} \\ +5 & \textrm{use one SPH factor assigned to the albedo function in each macro-group.} \\ +\end{array} \right. +\end{displaymath} + +\item The minimum group index $\mathcal{S}^{g}_{5}$, with $1 \le \mathcal{S}^{g}_{5}\le \mathcal{S}^{g}_{1}$. +\item The maximum group index $\mathcal{S}^{g}_{6}$, with $\mathcal{S}^{g}_{5} \le \mathcal{S}^{g}_{6}\le \mathcal{S}^{g}_{2}$. +\item The minimum mixture index $\mathcal{S}^{g}_{7}$, with $1 \le \mathcal{S}^{g}_{7}\le \mathcal{S}^{g}_{2}$. +\item The maximum mixture index $\mathcal{S}^{g}_{8}$, with $\mathcal{S}^{g}_{7} \le \mathcal{S}^{g}_{8}\le \mathcal{S}^{g}_{2}$. +\item The number of physical albedos $\mathcal{S}^{g}_{9}$. + +\end{itemize} +\goodbreak + +\subsubsection{The sub-directory /OLD-VALUE/ in /optimize/} + +\begin{DescriptionEnregistrement}{Main records and sub-directories in \dir{/OLD-VALUE/}}{7.0cm} +\DbleEnr +{VAR-VALUE\blank{3}}{$N_{var}$}{} +{The values of the decision variables of the last valid iteration.} + +\DbleEnr +{FOBJ-CST-VAL}{$N_{cst}+1$}{} +{The values of the objective function and the contraints of the last valid iteration.} + +\DbleEnr +{GRADIENT\blank{4}}{$N_{var}, N_{cst}+1$}{} +{The gradients of the objective function and the constraints of the last valid iteration.} + +\OptDbleEnr +{VAR-VALUE2\blank{2}}{$N_{var}$}{$\mathcal{S}^{o}_{8}=1$}{} +{The values of the decision variables of the second-last valid iteration.} + +\OptDbleEnr +{BEST-VAR\blank{4}}{$N_{var}$}{$\mathcal{S}^{o}_{8}=1$}{} +{The values of the decision variables corresponding to the best valid solution ever found.} + +\OptDbleEnr +{BEST-FCT\blank{4}}{1}{$\mathcal{S}^{o}_{8}=1$}{} +{The value of the objective function corresponding to the best valid solution ever found.} + +\end{DescriptionEnregistrement} +\clearpage + -- cgit v1.2.3